On the metric structure of section ring

The main goal of this article is to study for a projective manifold and an ample line bundle over it the relation between metric and algebraic structures on the associated section ring. More precisely, we prove that once the kernel is factored out, the multiplication operator of the section ring becomes an approximate isometry (up to normalization) with respect to the $L^2$-norms and the induced Hermitian tensor product norm. We also show that the analogous result holds for the $L^1$ and $L^{\infty}$-norms if instead of the Hermitian tensor product norm, we consider the projective and injective tensor norms induced by $L^1$ and $L^{\infty}$-norms respectively. Then we show that $L^2$-norms associated with continuous plurisubharmonic metrics are actually characterized by the multiplicativity properties of this type. Using this, we refine the theorem of Phong-Sturm about quantization of Mabuchi geodesics from the weaker level of Fubini-Study convergence to the stronger level of norm equivalences.